Here \(\sigma (\eta )\ge 0\)  is the spectral scaling function characterising the Lax spectrum of the soliton gas while \(\Gamma \)  is the support of \(f(\eta )\)  and \(v(\eta )\) .
Eliminating \(\sigma \)  one obtains the equation of state
(REF ) for the effective velocity of a “tracer” soliton in the gas, \(s(\eta )=v(\eta )/f(\eta )\) . The derivation of the equation (REF ) via the thermodynamic limit thus provides the justification of the collision rate assumption (REF ) for the KdV soliton gas.
The continuity (kinetic) equation (REF ) for slowly varying spectral DOS \(f(\eta , x, t)\)  in a non-equilibrium (weakly non-unifiorm) soliton gas is obtained as a thermodynamic limit of kinematic modulation equations expressing the “conservation of waves” [1]}, [2]}. A similar derivation has been recently performed in [3]} for the focusing NLS equation where the spectral parameter \(\lambda \)  is complex.
